To analyse the temporal evolution of the geometry of the network of veins and eventually its dynamical adaptation, given prescribed sources and sinks fluxes, the function \(g\)  in (REF ) must be chosen. For that, we introduce the simple criterion of minimisation of the dissipated energy per unit time during the flow
([1]}, [2]}, [3]}, [4]}), also preserving the total volume \({\cal V}\)  of fluid. So, by (REF ) and (REF ), we consider the Lagrangian
\({\cal L}={\cal P}-\lambda \left({\cal V}-\beta \sum _{(k,m)\in E}L_{km}\sqrt{D_{km}} \right),\) 
